3
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann3 Proof-Based Geometry Pythagoras’ Theorem: “The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse”. Already known to the Babylonians and Egyptians as experimental fact. Pythagorean innovation: –A proof, independent of experimental numerical verification Pythagoras of Samos (582 BC to 507 BC)

4
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann4 Proof-Based Geometry Pythagoras’ Theorem: “The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse”.

6
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann6 Algorithmic Geometry Ancient example (ca BC BC): Problem 50: A circular field of diameter 9 has the same area as a square of side 8. „Subtract 1/9 of the diameter which leaves 8 khet. The area is 8 multiplied by 8 or 64 setat“ Problem 48: Gives a hint of how this formula is constructed. Rhind Mathematical Papyrus (Ancient Egypt, ca BC)

7
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann7 Algorithmic Approach to Geometry 89 Problem: A circular field has diameter 9 khet. What is ist area? Solution: Subtract 1/9-th of the diameter which leaves 8 khet. The area is 8 multiplied by 8, or 64 setat.

9
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann9 Algorithmic Approach to Geometry Ancient method led to a very close approximate of the value PI (  ); up to 2% precision. Realises the “experimental quadrature of the circle”

10
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann10 Axiomatic Geometry Fundamental notions: –Points, straight lines, planes, incidence relation (“lies on”, “goes through”) A1: For any two points P and Q, there is exactly one straight line g on which both P and Q lie. A2: For each straight line g, there is one point which is not on g. Euclid of Alexandria (ca. 325 BC – 265 BC)

11
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann11 The Parallel Axiom A3: For each straight line g and each point P, which is not on g, there is exactly one straight line h, on which P lies and which does not have a common point with g. Question: Is A3 independent of A1 and A2?  Approach: Klein’s Model p h2 h1 g

14
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann14 The Geometry in GPS Technology The process of trilateration (similar to triangulation) with at least 3 satellites. Fourth satellite is used to synchronise time signals.

16
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann16 Huzita’s Axioms A1: Given two points p1 and p2, there is a unique fold that passes through both of them. A2: Given two points p1 and p2, there is a unique fold that places p1 onto p2. A3: Given two lines l1 and l2, there is a fold that places l1 onto l2. A4: Given a point p1 and a line l1, there is a unique fold perpendicular to l1 that passes through point p1.

17
Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann17 Huzita’s Axioms A5: Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2. A6: Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2. A7: Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and perpendicular to l2. Geometry based on these axioms is more powerful than the standard Compass-and-straightedge Geometry!